Largest Component in Boolean Sublattices

 For a subfamily F ⊆ 2[ⁿ] of the Boolean lattice, consider the graph G_F on F based on the pairwise inclusion relations among its members. Given a positive integer t, how large can F be before G_F must contain some com ponent of order greater than t? For t = 1, this question was answered exactly almost a century ago by Sperner: the size of a middle layer of the Boolean lattice. For t = 2ⁿ, this question is trivial. We are interested in what happens between these two extremes. For t = 2^g with g = g(n) being any integer function that satisfies g(n) = o(n/log n) as n → ∞, we give an asymptotically sharp answer to the above question: not much larger than the size of a middle layer. This constitutes a nontrivial generalisation of Sperner’s theorem. We do so by a reduction to a Tur´an-type problem for rainbow cycles in properly edge-coloured graphs. Among other results, we also give a sharp answer to the question, how large can F be before G_F must be connected?